Plasma Astrophysics Chapter 7-2: Instabilities II Yosuke Mizuno Institute of Astronomy National Tsing-Hua University

Plasma AstrophysicsChapter 7-2: Instabilities II

Yosuke Mizuno

Institute of Astronomy

National Tsing-Hua University

Table of contents

(last week)• What is the instability? (How to analyze the instability)• Rayleigh-Taylor instability• Kelvin-Helmholtz instability

(today)• Parker instability• Magneto-rotational instability• Jeans instability• Current-driven instability

Magnetic buoyancy

• Convective motion in fluid is driven by thermal buoyancy• In compressible plasma, we have another important buoyancy, i.e.,

magnetic buoyancy

• Consider isolated flux tube (density ri, gas pressure pi and magnetic pressure pm) embedded in nonmagnetized plasma (density re, gas pressure pe ) under uniform gravity g

• Assume: isothermal, i.e., the temperature Ti = Te= T

• Assume: tube is thin, i.e., the radius of flux tube is much smaller than local pressure scale height

Magnetic buoyancy (cont.)• Equilibrium configuration of magnetic flux tube is determined by

the balance of total pressure

• Then, from EoS of p = rRT, the density inside the tube (ri) becomes smaller than the density outside the tube (re)

• Therefore

• Hence the tube suffers the buoyancy force

• This is called magnetic buoyancy• This is fundamental force to raise the flux tube to the surface of the

Sun, stars and accretion disk (galactic gas disk)

Magnetic buoyancy (cont.)• A horizontal isothermal, isolated flux tube cannot be in equilibrium.• On the other hand, a 2D isothermal flux sheet can be in equilibrium• Even in this case, the sheet often becomes unstable because of

magnetic buoyancy• There are two kinds of magnetic buoyancy instability • Interchange mode : – Necessary condition: – wavelength: arbitrary l– Flute instability, magnetic RT instability (Kruskal-Schwarzschild

instability)• Undular mode:– Necessary condition:

– Wavelength: l > lc ~ 10H

– Ballooning instability, Parker instability

Magnetic buoyancy (cont.)

flux sheet

Interchange modek \perp B

Undular modek || B

Parker instability

• Parker (1966) emphasized an importance of magnetic buoyancy instability (undular mode) in the Galactic disk (including cosmic-ray pressure effect) and explain the formation of interstellar cloud complexes.

• Hence, in astrophysics, magnetic buoyancy instability (undular mode) is usually called Parker instability

Parker instability (cont.)

Magnetic field lift-up from equilibrium state

Plasma falls down along bending magnetic field lines

Top region becomes more lighter. Then buoyancy force is working more (magnetic field lift-up more)= growth of instability

g

B


• Here, drive instability condition• Consider : a horizontal flux sheet in magneto-static equilibrium with

gravity ( and )

• Assume: isothermal and plasma beta b= 2m0p /B2 = constant

• From isothermal condition (assume g=1),

Parker instability (cont.)• From pressure balance (to gravity) in z-direction,

• When there is no magnetic field (b=0), • : hydrostatic equilibrium• where H is scale height:• When the magnetic field is exist,

• Where• In the magnetic field supported disk, the plasma is located higher

region than hydrostatic disk case.


• From equilibrium state, magnetic field lifts up Dz (<< 1).• Plasma lift-up time-scale is much longer than sound crossing time

scale. So maintains pressure balance everywhere. • After lift-up, plasma inside the bent magnetic field can move along

field lines. Therefore magnetic pressure inside bent magnetic field does not change.

g

Parker instability (cont.)• Calculate density variance between the inside and outside of the bent

magnetic field

• It shows that inside the bent magnetic field becomes lighter• Plasma inside the bent magnetic field has buoyancy force (+z

direction)

• On the other hand, due to bent magnetic field, there is magnetic tension force in –z direction

(7.43)


• Here curvature radius is R, the magnetic tension is estimated as

• Consider triangle in the circle R, the relation between R and Dz is

• Here l is perturbed wavelength (distance AB ~ l/4)

(7.44)

(7.45)

Parker instability (cont.)• If Fbuoyancy > Ftension, plasma inside the bent magnetic field

continuously lift-up. It means that it is unstable

• From eq (7.43), (7.44), (7.45), critical wavelength is

• Therefore if the magnetic field is perturbed the wavelength enough longer than scale-height, the buoyancy overcomes magnetic tension and gravity then inside plasma continuously lift-up (unstable).

• This is so-called Parker instability• Growth rate is roughly estimated by


Solar coronal loop （ Three year obs., by SDO）

Molecular Loops in the Galactic Center (radio CO obs)

Movie here

Parker instability (cont.)Movie

2D MHD simulations of Parker Instability

http://www.astro.phys.s.chiba-u.ac.jp/netlab/cans/cans2d/movie/frame.html

Magneto-rotational instability• Important for angular momentum transport in accretion disk• In the standard theory of accretion disks (Shakura & Sunyaev

1973), the a-prescription of viscosity is adopted for radial angular momentum transport

• What phenomenological viscosity parameter a?• From observation of dwarf novae, a=0.02 (quiescent) - 0.1

(bursting phase)• Molecular viscosity: NO (too small)• Hydrodynamic shear flow instability makes convective turbulence

in accretion disk (turbulent viscosity). – But in geometrically thin Keplerian disk, a=O(10-3)

• In MHD model: magnetic stress enhanced turbulent viscosity incurred by fluctuating magnetic field <= generated by Magnetorotational instability (MRI) (Balbus & Hawley 1991)

Magneto-rotational instability (cont.)

• Understanding of MRI through Lagrangian point of view• Consider differentially rotating plasma disk with vertical magnetic

field (penetrate disk) in some gravitational field (stationary)• Put small radial perturbation in rotating plasma at radius r from

rotation axis (angular momentum is conserved) and moves to r+Dr • The angular velocity in r+Dr is slower than that in r. Thus magnetic

field is deformed more and magnetic tension is happened.

Side view top view

r r +Dr


• Due to the magnetic tension, plasma is accelerated to rotational direction.

• The plasma in r+Dr tries to rotate with angular velocity at r.• This faster angular velocity makes stronger centrifugal force which

is larger than gravitational force.• Then the plasma is push outward more. Again magnetic field is

stretched more and make larger magnetic tension.• This process is so-called Magneto-rotational instability (MRI).

acceleration

Magnetic tension Stronger

centrifugal force

Magneto-rotational instability (cont.)• Rough estimate the instability condition • Assume Keplerian rotating plasma disk with vertical magnetic field

(penetrate disk) in some gravitational field (stationary)• Put small radial perturbation in rotating plasma at radius r from

rotation axis (angular momentum is conserved) and moves to r+Dr • Consider radial force balance at r+Dr– Gravity:

– Centrifugal force:

• Where, the effect of acceleration by magnetic tension in rotational direction is included in centrifugal force


• From Keplerian rotation,

• Radial force including gravity and centrifugal force is

• Next calculate radial force by magnetic tension.• As shown in figure, the deformation of magnetic field is

approximate as a circle with radius x• Magnetic tension is

Magneto-rotational instability (cont.)• From similarity relation

• Using this value, magnetic tension is

• The system is unstable when (gravity + centrifugal force) > (magnetic tension in radial direction)

• Therefore, the condition for growing instability is

=>

Magneto-rotational instability (cont.)• From instability condition, the instability occurs longer than the

critical wavelength.

• This feature is similar to that of Parker instability, i.e., stabilized by magnetic tension force.

• If magnetic field is strong, this instability is stabilized because the critical wavelength lc exceeds the disk thickness H.

• In this case, the critical field strength for stability is

• For growth of MRI, weak magnetic field in the accretion disk is important


• Next, we derive dispersion relation of MRI,• Linearized equations


• Consider the frame of rotating around z-axis with angular velocity W(r) in cylindrical coordinates (r, f, z) i.e.,

• In equilibrium state, gravity and centrifugal force is balanced• Uniform magnetic field,• Perturbation: , wavenumber• In detail of calculation, need to use from local

analysis • After some manipulations, we get following dispersion relation (for

simply use B0f=0),

Magneto-rotational instability (cont.)• Rigid rotation case

– From , the dispersion relation is

– There is no solution with w2 < 0, therefore rigid rotation disk is stable against MRI

• Keplerian rotation case

– From this, the dispersion relation is

– When , w2< 0. Therefore Keplerian disk is unstable against MRI. And maximum growth rate is


• In Keplerian disk, growth rate is comparable with W, i.e., this instability is fairly fast instability occurring at the rotation time scale of disk

• This instability occurs even if the magnetic field is very weak• People often neglected the effect of magnetic field in accretion

disk simply because magnetic field is very weak in the disk• But from properties of MRI, we cannot neglect magnetic field

any more.


3D MHD Simulation in local shearing box

3D MHD in global accretion disk

Movie here

Movie here

Movie here

Jeans instability• In many astrophysical phenomena, gravitational field plays an

important role.• In particular, self-gravity and the associated instability are essential

when we consider the formation of various objects (e.g., stars, galaxies, and the clusters of galaxies) due to density fluctuations

• Consider an infinite homogeneous medium at rest r =r0=uniform, p=p0=uniform, v=v0=0, F=F0

• Here we consider self-gravity of medium but neglect magnetic field and assume adiabatic, p=Krg

Jeans instability (cont.)• Linearized equations

• From these equations, we obtain

• If G=0, this equation expresses the propagation of sound wave in a homogeneous medium.

• In other word, this equation shows that how the propagation of sound wave is modified in self-gravity field

Jeans instability (cont.)• If we consider a plane wave and put

we obtain the dispersion relation

• w2 becomes negative when

• where lJ is so-called Jeans wavelength (radius).

• In the perturbation with longer wavelength, attracting force from self-gravity overcomes increase of gas pressure then gravitational collapse is occurred (unstable)

• This is so-called Jeans instability.

Jeans instability (cont.)• Compute the mass contained within the Jeans radius (consider as

a sphere)

• Here MJ is so-called Jeans Mass. From this instability, the object with M > MJ is formed.

• Jeans mass is small if temperature is low and density is high.• Therefore because of the density increase by the cloud contracts,

the Jeans wavelength becomes shorter and shorter. • It means that the Jeans instability is takes place at smaller and

smaller scales as the cloud contracts, leading to a fragmentation into many small pieces.

Jeans instability (cont.)

• In this lecture, we consider the simple model, an infinite homogeneous medium at rest

• If we consider the rotating disk, the Coriolis force is protected to contraction of gas => instability condition is changing (Toomre 1964)

Jeans instability (cont.)

Star formation in molecular cloud

3D SPH simulations of star formation from gas cloud

Movie here

Current driven (kink) instability

• A linear pinched discharge in the laboratory is a cylindrical plasma column (radius a) that is confined (or pinched) by toroidal magnetic field due to current ( ) flowing along its surface or through its interior

• This configuration is similar to magnetic flux tubes present in the solar atmosphere and astrophysical jets formed from compact rotator.

• So summaries its stability properties here

a

Bf

rJ

Current driven (kink) instability (cont.)

• The radially inwards J x B force (magnetic pressure B2/2m0 and magnetic tension B2/2m0r) is balanced by outwards pressure gradient.

• When plasma (at pressure p0 & density r0) contains no magnetic field (interior), the pinch is unstable to the interchange mode (k \perp B), since the confining field is concave to plasma

• The place where it pinched, toroidal field is increases and radius is deceases. Therefore magnetic pressure and tension increase => inward force is no longer balances with gas pressure => perturbation grows

• The place where it bulges out, toroidal field is decreases and radius is increases. Therefore magnetic pressure and tension decrease => perturbation grows

Bf

Current driven (kink) instability (cont.)• This instability is so-called sausage instability (m=0 mode of

current-driven instability, sausage mode).• This instability is unstable in all wavelength (for cylindrical plasma

column with toroidal field)• The growth rate of this instability with the wavenumber

• The cylindrical plasma column can be stabilized against the sausage mode by the presence of a large enough axial field (B0z)

• The value of toroidal field at the interface is Bf, the force balance on the interface gives

Current driven (kink) instability (cont.)• The effect of Alfven wave propagating along the axis with speed

is to modify the dispersion relation to

• This force balance gives stability (w2 > 0) when


• Consider the perturbation of kink to cylindrical plasma column

• Inside kinked plasma column, magnetic pressure becomes strong, while outside of the kinked plasma column, magnetic pressure becomes weak => perturbation grows (unstable).

• This instability is so-called kink instability (m=1 mode of current-driven instability, sausage mode)

• The axial field in cylindrical plasma column also affects the stabilize of this instability (kink-mode)

a


• The condition of stability for kink mode is

• If the perturbed wavelength is long enough, the plasma column with helical magnetic field is unstable against kink instability.

Kruskal-Shafranov criterion


Sausage pinch instability in solar corona (Obs by SDO)

Kink instability in laboratory experiment

MHD simulation of kink instability

Movie here

Summary• There are many potentially growing instabilities in the universe.• These instabilities are strongly related the dynamics in the

universe.• Important: – what system is stable/unstable against instabilities (condition

for stable/unstable of instability)– What is the time scale of growing instabilities (growth rate).

Does it affects the dynamics of system?• Here not covered…but may be important

– Thermal instability, radiation (pressure)-driven instability, Richitmeyer-Meshkov instability, Corrugation instability, Tearing instability …

Documents

Plasma Astrophysics Chapter 7-2: Instabilities II Yosuke Mizuno Institute of Astronomy National Tsing-Hua University